What Groupon wants doesn't particularly matter to the situation. Unless Groupon has somehow convinced you to sign a contract promising that you'll never give anyone a better discount than the Groupon, you can handle regular customers however you want - including customers who aren't doing business under Groupon terms, because their Groupon is expired and doesn't apply. So: "Sure, Lucy. Here you go."

Now, there is a real barrier to just giving Lucy the deal. The entire point of the blog post is that the discount was ruinously unaffordable and didn't attract customers. When you can't make payroll, maybe you genuinely can't afford to comp six dollars of flavored water! Or maybe you don't want to set the public precedent of honoring expired coupons when the coupons were bad business even during their active dates and an entire line of customers is listening to this exchange while waiting behind Lucy. Maybe you get a flood of people digging out their expired coupons, or just demanding to be treated as well as Lucy - exactly what you can't afford. If something is unsustainable then you need to stop sustaining it, and that means there has to be a first person you tell "No."

I suspect that was the sense in which the original poster said she "can't" give Lucy the deal: The deal can't be continued - it's been a disaster - it can't go on. Strictly speaking, you have the power to treat customers however you want, but at the end of a bad business decision related to customer discounts, it's only human to frame everything in terms of "we need to get out of this bad deal."

But taking the Groupon deal at a bad price, such that she painted herself into a corner with her best customer, was still the original poster's own mistake.

I feel it would be unambiguously better, in a social and moral sense, if Groupon provided more advice and guidance to discount providers so that they don't risk going out of business. The mistakes may have been the responsibility of Posie's Cafe, but problems are problems, and Groupon wasn't exactly an uninterested bystander in the issue. I think it would probably be better for Groupon's long-term economic interests as well; if you burn your business partners, you earn a bad reputation, and eventually you run out of people who haven't heard the nasty rumors.